The Schur basis on n d-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. I will present efficient (poly(n,d,log 1/epsilon) size for accuracy epsilon) circuits quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. These circuits are based on efficient circuits for the Clebsch-Gordan transformation.
I will also present an efficient circuit for a limited version of the Schur transform in which one needs only to project onto different Schur subspaces. This circuit can then be generalized to a generic circuit for performing the change of basis from a standard basis into a basis in which the action of a particular representation of a finite group is block diagonal and irreducible. This construction generalizes phase estimation from an abelian setting to a generic setting involving nonabelian finite groups.